Structured Sets CS1200, CSE IIT Madras Meghana Nasre April 27, - PowerPoint PPT Presentation

Structured Sets CS1200, CSE IIT Madras Meghana Nasre April 27, 2020 CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Structured Sets • Relational Structures • Properties and closures � • Equivalence Relations � • Partially Ordered Sets (Posets) and Lattices � • Algebraic Structures • Groups and Rings CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Algebraic Structures: Recap Set A with a binary operator ∗ • If ∗ is closed and associative, and an identity element e exists, and every element b ∈ A has an inverse then ( A , ∗ ) is a group. CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Algebraic Structures: Recap Set A with a binary operator ∗ • If ∗ is closed and associative, and an identity element e exists, and every element b ∈ A has an inverse then ( A , ∗ ) is a group. • If B ⊆ A and ( B , ∗ ) forms a group, then B is a sub-group of ( A , ∗ ). CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Algebraic Structures: Recap Set A with a binary operator ∗ • If ∗ is closed and associative, and an identity element e exists, and every element b ∈ A has an inverse then ( A , ∗ ) is a group. • If B ⊆ A and ( B , ∗ ) forms a group, then B is a sub-group of ( A , ∗ ). • Generator of a group and cyclic groups. CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Algebraic Structures: Recap Set A with a binary operator ∗ • If ∗ is closed and associative, and an identity element e exists, and every element b ∈ A has an inverse then ( A , ∗ ) is a group. • If B ⊆ A and ( B , ∗ ) forms a group, then B is a sub-group of ( A , ∗ ). • Generator of a group and cyclic groups. Example group that is not cyclic. * a b c d a a b c d b b a d c c c d a b d d c b a CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Algebraic Structures: Recap Set A with a binary operator ∗ • If ∗ is closed and associative, and an identity element e exists, and every element b ∈ A has an inverse then ( A , ∗ ) is a group. • If B ⊆ A and ( B , ∗ ) forms a group, then B is a sub-group of ( A , ∗ ). • Generator of a group and cyclic groups. Example group that is not cyclic. * a b c d a a b c d b b a d c c c d a b d d c b a • Lagrange’s Theorem: The order of any subgroup of a finite group divides the order of the group. CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . B 1 = { 1 , 3 , 5 } B 2 = { 2 , 4 , 0 } B 3 = { 3 , 5 , 1 } CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . B 1 = { 1 , 3 , 5 } B 2 = { 2 , 4 , 0 } B 3 = { 3 , 5 , 1 } CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . B 1 = { 1 , 3 , 5 } B 2 = { 2 , 4 , 0 } B 3 = { 3 , 5 , 1 } Observe the difference between the cosets obtained when the subset forms a subgroup CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . B 1 = { 1 , 3 , 5 } B 2 = { 2 , 4 , 0 } B 3 = { 3 , 5 , 1 } Observe the difference between the cosets obtained when the subset forms a subgroup (recall B , ⊕ 6 ) is a group, CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Example: Z 6 = { 0 , 1 , 2 , 3 , 4 , 5 } ( Z 6 , ⊕ 6 ) is a group. Consider the subset H = { 0 , 1 , 5 } . H 1 = { 1 , 2 , 0 } H 2 = { 2 , 3 , 1 } H 3 = { 3 , 4 , 2 } Now let us consider a set B = { 0 , 2 , 4 } . B 1 = { 1 , 3 , 5 } B 2 = { 2 , 4 , 0 } B 3 = { 3 , 5 , 1 } Observe the difference between the cosets obtained when the subset forms a subgroup (recall B , ⊕ 6 ) is a group, whereas ( H , ⊕ 6 ) is not a group. CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Claim: If ( H , ∗ ) is a subgroup of ( A , ∗ ) then for any c ∈ A and d ∈ A , either H c = H d or H c ∩ H d = ∅ . CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Claim: If ( H , ∗ ) is a subgroup of ( A , ∗ ) then for any c ∈ A and d ∈ A , either H c = H d or H c ∩ H d = ∅ . Proof: Let H c ∩ H d � = ∅ . Let f ∈ H c ∩ H d . CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Claim: If ( H , ∗ ) is a subgroup of ( A , ∗ ) then for any c ∈ A and d ∈ A , either H c = H d or H c ∩ H d = ∅ . Proof: Let H c ∩ H d � = ∅ . Let f ∈ H c ∩ H d . Thus there exists h 1 and h 2 in H such that f = c ∗ h 1 = d ∗ h 2 . CS1200, CSE IIT Madras Meghana Nasre Structured Sets

Cosets of a subset Let ( A , ∗ ) be a group and H be any subset of A . For any element c ∈ A , the left coset of H w.r.t. c is defined as: H c = { c ∗ b | b ∈ H } Claim: If ( H , ∗ ) is a subgroup of ( A , ∗ ) then for any c ∈ A and d ∈ A , either H c = H d or H c ∩ H d = ∅ . Proof: Let H c ∩ H d � = ∅ . Let f ∈ H c ∩ H d . Thus there exists h 1 and h 2 in H such that f = c ∗ h 1 = d ∗ h 2 . Since ( H , ∗ ) is a group, inverse exists for every element, in particular h 1 . Therefore c = d ∗ h 2 ∗ h − 1 1 . CS1200, CSE IIT Madras Meghana Nasre Structured Sets